Now, it seems that we can start to perform numerical
simulations. This is true, but we will not obtain acceptable results without knowing something
about computation schemes.

Computation schemes are the ways that we use to push the evolution
of the system. Different schemes are developed to overcome certain numerical difficulties that
we meet in numerical simulation. Here we choose a typical one dimensional fluid dynamics
equation as an example.

(61)

where u is the system variable, f is the flux.

The naive way to differentiate this equation is

(62)

where uni is the u value of space position i, time n. From
this we can get the evolution of the variable u

(63)

Though this naive way is the most natural way to compute the time evolution of
simulation system, later we will see that it is far from satisfactory in
a practical simulation. Following are some other schemes developed to overcome
the numerical difficulty of the naive scheme.

First order scheme: Rusanov scheme

(fni+1-fni-1)

(64)

Second order scheme: Lax-Wendroff scheme

(65)

(66)

Other schemes, such as upwind, MacCormack, hybrid, etc. are also common
schemes in numerical simulation.